Neither of the two cases are completely inconceivable:
A homogeneous, anisotropic universe
A universe with galaxies spread evenly all over, but all spinning in the same direction. This universe would look the same no matter where you lived, but have a net angular momentum, so looking in one direction you'd see all galaxies spinning along your line of sight, and in another direction, you'd see them spinning perpendicular to this direction.
Another example is a universe that had been permeated by density waves in one direction. In this direction, you'd see the density of galaxies alternating between high and low, and perpendicular hereto you'd see a constant density.
Yesterday's papers on arXiv included a paper (Schucker 2016) that discusses the the possibility that we might live in another type of homogeneous, anisotropic universe, namely one in which the observed expansion rate depends upon the direction in which you look. This is called a "Bianchi I universe", and isn't just a hypothetical curiosity (although the results of this paper is statisically non-significant). See also @JonesTheAstronomer's answer.
An inhomogeneous, isotropic universe
As John Rennie has taught us, Big Bang didn't happen at a point. However, if it did, and we happened to live in the central region, we could observe the same in all directions, but see a gradually thinnening universe, or maybe increasing to some point and then decreasing, depending on exactly how this exsplosion came about. This scenario would however imply that we inhabit a special place in the universe, which would make Kopernikus sad. If a universe is isotropic from more than one location, is must also be homogeneous.
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